In this lecture we discuss how to calculate the inverse Laplace transform. The simplest way to invert a Laplace transform is to convert the function to be inverted into a sum of functions which are Laplace transforms that we have already calculated. By the linearity of Laplace transforms, the given function can be inverted directly and its inverse will be a sum of the corresponding time domain functions. The table given under the supplementary material in this web site will be very useful in this. It lists some common time domain functions and their Laplace transforms. Some of these relationships were derived in Lecture 6.

We will next give an example that demonstrates how the Laplace transform can be used in solving an ordinary differential equation. Note that the inverse Laplace transform will be needed to solve the problem.

Example 1:

The differential equation of a system is given as

where u is the input to the system and x is the output of the system. Find the output signal x(t) if the input

Take Laplace transform of both sides of the differential equation

In the following we will study by examples the techniques of inverting Laplace transforms. Each example will represent a particular form for a Laplace transform, and the solution can be viewed as a model on which to base the procedure for similar cases. Note that in the previous example we provided a method for inverting the Laplace transform.

Example 2: Calculate the inverse Laplace transform for the following function

Using the partial fraction expansion, we have

Example 3: Calculate the inverse Laplace transform for the following function

Using the partial fraction expansion, we have

Example 4: Calculate the inverse Laplace Transform for the following function. Note that the denominator has real and complex roots.

The roots of the denominator of X(s) are s = -6, s = -3+4j and s = -3-4j . We first calculate the partial fraction expansion of X(s) without factoring the second order polynomial which has the complex roots.